116 ELECTRICAL MEASUREMENTS 



of the galvanometer. If e goes through its cycle " instanta- 

 neously," that is, in a time so short that the cycle is over before 

 the coil of the galvanometer moves appreciably, then as the 

 current is zero both at the start and at the finish, 



ft (*5 t' 



- 1 idt = I edt = rQ. 



Jo Jo 



In the case of an instantaneous displacement of electricity 

 through the instrument the quantity, Q, is given by 



(23) 

 tVTT'-f-X* j 



This is the formula commonly used for the ballistic galvanometer. 

 The term involving X gives the correction for damping. 



The quantity in the brackets { } may be expanded by 

 Maclaurin's theorem, giving 



. : vV + x 2 



and if X is small, 



.= 1 + Q.5X - 0.026X 2 - 0.055X* .. 



For a secondary instrument, assuming that T is not sensibly 

 affected by any change in damping which is likely to be encoun- 

 tered, 



(25) 

 where K is a constant. 



Another approximation sometimes used in making the correc- 

 tion for damping, results from assuming X to be very small, in 

 which case 



7T X tan-^ 



~~/~~ 2 ~ 2 ** x = * 2 = 2 approximately. 



ir I* "\ *\ 



But, from the law of damped oscillations (page 30), 



02 03 



Consequently 



="=" 



